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The nonlinear behavior of concrete can be attributed to the propagation of microcracks within the heterogeneous internal material structure. In this thesis, a mesoscale model is developed which allows for the explicit simulation of these microcracks. Consequently, the actual physical phenomena causing the complex nonlinear macroscopic behavior of concrete can be represented using rather simple material formulations. On the mesoscale, the numerical model explicitly resolves the components of the internal material structure. For concrete, a three-phase model consisting of aggregates, mortar matrix and interfacial transition zone is proposed. Based on prescribed grading curves, an efficient algorithm for the generation of three-dimensional aggregate distributions using ellipsoids is presented. In the numerical model, tensile failure of the mortar matrix is described using a continuum damage approach. In order to reduce spurious mesh sensitivities, introduced by the softening behavior of the matrix material, nonlocal integral-type material formulations are applied. The propagation of cracks at the interface between aggregates and mortar matrix is represented in a discrete way using a cohesive crack approach. The iterative solution procedure is stabilized using a new path following constraint within the framework of load-displacement-constraint methods which allows for an efficient representation of snap-back phenomena. In several examples, the influence of the randomly generated heterogeneous material structure on the stochastic scatter of the results is analyzed. Furthermore, the ability of mesoscale models to represent size effects is investigated. Mesoscale simulations require the discretization of the internal material structure. Compared to simulations on the macroscale, the numerical effort and the memory demand increases dramatically. Due to the complexity of the numerical model, mesoscale simulations are, in general, limited to small specimens. In this thesis, an adaptive heterogeneous multiscale approach is presented which allows for the incorporation of mesoscale models within nonlinear simulations of concrete structures. In heterogeneous multiscale models, only critical regions, i.e. regions in which damage develops, are resolved on the mesoscale, whereas undamaged or sparsely damage regions are modeled on the macroscale. A crucial point in simulations with heterogeneous multiscale models is the coupling of sub-domains discretized on different length scales. The sub-domains differ not only in the size of the finite elements but also in the constitutive description. In this thesis, different methods for the coupling of non-matching discretizations - constraint equations, the mortar method and the arlequin method - are investigated and the application to heterogeneous multiscale models is presented. Another important point is the detection of critical regions. An adaptive solution procedure allowing the transfer of macroscale sub-domains to the mesoscale is proposed. In this context, several indicators which trigger the model adaptation are introduced. Finally, the application of the proposed adaptive heterogeneous multiscale approach in nonlinear simulations of concrete structures is presented.
Briefly, the two basic questions that this research is supposed to answer are:
1. Howmuch fiber is needed and how fibers should be distributed through a fiber reinforced composite (FRC) structure in order to obtain the optimal and reliable structural response?
2. How do uncertainties influence the optimization results and reliability of the structure?
Giving answer to the above questions a double stage sequential optimization algorithm for finding the optimal content of short fiber reinforcements and their distribution in the composite structure, considering uncertain design parameters, is presented. In the first stage, the optimal amount of short fibers in a FRC structure with uniformly distributed fibers is conducted in the framework of a Reliability Based Design Optimization (RBDO) problem. Presented model considers material, structural and modeling uncertainties. In the second stage, the fiber distribution optimization (with the aim to further increase in structural reliability) is performed by defining a fiber distribution function through a Non-Uniform Rational BSpline (NURBS) surface. The advantages of using the NURBS surface as a fiber distribution function include: using the same data set for the optimization and analysis; high convergence rate due to the smoothness of the NURBS; mesh independency of the optimal layout; no need for any post processing technique and its non-heuristic nature. The output of stage 1 (the optimal fiber content for homogeneously distributed fibers) is considered as the input of stage 2. The output of stage 2 is the Reliability Index (b ) of the structure with the optimal fiber content and distribution.
First order reliability method (in order to approximate the limit state function) as well as different material models including Rule of Mixtures, Mori-Tanaka, energy-based approach and stochastic multi-scales are implemented in different examples. The proposed combined model is able to capture the role of available uncertainties in FRC structures through a computationally efficient algorithm using all sequential, NURBS and sensitivity based techniques. The methodology is successfully implemented for interfacial shear stress optimization in sandwich beams and also for optimization of the internal cooling channels in a ceramic matrix composite.
Finally, after some changes and modifications by combining Isogeometric Analysis, level set and point wise density mapping techniques, the computational framework is extended for topology optimization of piezoelectric / flexoelectric materials.
This thesis presents the advances and applications of phase field modeling in fracture analysis. In this approach, the sharp crack surface topology in a solid is approximated by a diffusive crack zone governed by a scalar auxiliary variable. The uniqueness of phase field modeling is that the crack paths are automatically determined as part of the solution and no interface tracking is required. The damage parameter varies continuously over the domain. But this flexibility comes with associated difficulties: (1) a very fine spatial discretization is required to represent sharp local gradients correctly; (2) fine discretization results in high computational cost; (3) computation of higher-order derivatives for improved convergence rates and (4) curse of dimensionality in conventional numerical integration techniques. As a consequence, the practical applicability of phase field models is severely limited.
The research presented in this thesis addresses the difficulties of the conventional numerical integration techniques for phase field modeling in quasi-static brittle fracture analysis. The first method relies on polynomial splines over hierarchical T-meshes (PHT-splines) in the framework of isogeometric analysis (IGA). An adaptive h-refinement scheme is developed based on the variational energy formulation of phase field modeling. The fourth-order phase field model provides increased regularity in the exact solution of the phase field equation and improved convergence rates for numerical solutions on a coarser discretization, compared to the second-order model. However, second-order derivatives of the phase field are required in the fourth-order model. Hence, at least a minimum of C1 continuous basis functions are essential, which is achieved using hierarchical cubic B-splines in IGA. PHT-splines enable the refinement to remain local at singularities and high gradients, consequently reducing the computational cost greatly. Unfortunately, when modeling complex geometries, multiple parameter spaces (patches) are joined together to describe the physical domain and there is typically a loss of continuity at the patch boundaries. This decrease of smoothness is dictated by the geometry description, where C0 parameterizations are normally used to deal with kinks and corners in the domain. Hence, the application of the fourth-order model is severely restricted. To overcome the high computational cost for the second-order model, we develop a dual-mesh adaptive h-refinement approach. This approach uses a coarser discretization for the elastic field and a finer discretization for the phase field. Independent refinement strategies have been used for each field.
The next contribution is based on physics informed deep neural networks. The network is trained based on the minimization of the variational energy of the system described by general non-linear partial differential equations while respecting any given law of physics, hence the name physics informed neural network (PINN). The developed approach needs only a set of points to define the geometry, contrary to the conventional mesh-based discretization techniques. The concept of `transfer learning' is integrated with the developed PINN approach to improve the computational efficiency of the network at each displacement step. This approach allows a numerically stable crack growth even with larger displacement steps. An adaptive h-refinement scheme based on the generation of more quadrature points in the damage zone is developed in this framework. For all the developed methods, displacement-controlled loading is considered. The accuracy and the efficiency of both methods are studied numerically showing that the developed methods are powerful and computationally efficient tools for accurately predicting fractures.
Polymeric nanocomposites (PNCs) are considered for numerous nanotechnology such as: nano-biotechnology, nano-systems, nanoelectronics, and nano-structured materials. Commonly , they are formed by polymer (epoxy) matrix reinforced with a nanosized filler. The addition of rigid nanofillers to the epoxy matrix has offered great improvements in the fracture toughness without sacrificing other important thermo-mechanical properties. The physics of the fracture in PNCs is rather complicated and is influenced by different parameters. The presence of uncertainty in the predicted output is expected as a result of stochastic variance in the factors affecting the fracture mechanism. Consequently, evaluating the improved fracture toughness in PNCs is a challenging problem.
Artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) have been employed to predict the fracture energy of polymer/particle nanocomposites. The ANN and ANFIS models were constructed, trained, and tested based on a collection of 115 experimental datasets gathered from the literature. The performance evaluation indices of the developed ANN and ANFIS showed relatively small error, with high coefficients of determination (R2), and low root mean square error and mean absolute percentage error.
In the framework for uncertainty quantification of PNCs, a sensitivity analysis (SA) has been conducted to examine the influence of uncertain input parameters on the fracture toughness of polymer/clay nanocomposites (PNCs). The phase-field approach is employed to predict the macroscopic properties of the composite considering six uncertain input parameters. The efficiency, robustness, and repeatability are compared and evaluated comprehensively for five different SA methods.
The Bayesian method is applied to develop a methodology in order to evaluate the performance of different analytical models used in predicting the fracture toughness of polymeric particles nanocomposites. The developed method have considered the model and parameters uncertainties based on different reference data (experimental measurements) gained from the literature. Three analytical models differing in theory and assumptions were examined. The coefficients of variation of the model predictions to the measurements are calculated using the approximated optimal parameter sets. Then, the model selection probability is obtained with respect to the different reference data.
Stochastic finite element modeling is implemented to predict the fracture toughness of polymer/particle nanocomposites. For this purpose, 2D finite element model containing an epoxy matrix and rigid nanoparticles surrounded by an interphase zone is generated. The crack propagation is simulated by the cohesive segments method and phantom nodes. Considering the uncertainties in the input parameters, a polynomial chaos expansion (PCE) surrogate model is construed followed by a sensitivity analysis.
Encapsulation-based self-healing concrete (SHC) is the most promising technique for providing a self-healing mechanism to concrete. This is due to its capacity to heal fractures effectively without human interventions, extending the operational life and lowering maintenance costs. The healing mechanism is created by embedding capsules containing the healing agent inside the concrete. The healing agent will be released once the capsules are fractured and the healing occurs in the vicinity of the damaged part. The healing efficiency of the SHC is still not clear and depends on several factors; in the case of microcapsules SHC the fracture of microcapsules is the most important aspect to release the healing agents and hence heal the cracks. This study contributes to verifying the healing efficiency of SHC and the fracture mechanism of the microcapsules. Extended finite element method (XFEM) is a flexible, and powerful discrete crack method that allows crack propagation without the requirement for re-meshing and has been shown high accuracy for modeling fracture in concrete. In this thesis, a computational fracture modeling approach of Encapsulation-based SHC is proposed based on the XFEM and cohesive surface technique (CS) to study the healing efficiency and the potential of fracture and debonding of the microcapsules or the solidified healing agents from the concrete matrix as well. The concrete matrix and a microcapsule shell both are modeled by the XFEM and combined together by CS. The effects of the healed-crack length, the interfacial fracture properties, and microcapsule size on the load carrying capability and fracture pattern of the SHC have been studied. The obtained results are compared to those obtained from the zero thickness cohesive element approach to demonstrate the significant accuracy and the validity of the proposed simulation. The present fracture simulation is developed to study the influence of the capsular clustering on the fracture mechanism by varying the contact surface area of the CS between the microcapsule shell and the concrete matrix. The proposed fracture simulation is expanded to 3D simulations to validate the 2D computational simulations and to estimate the accuracy difference ratio between 2D and 3D simulations. In addition, a proposed design method is developed to design the size of the microcapsules consideration of a sufficient volume of healing agent to heal the expected crack width. This method is based on the configuration of the unit cell (UC), Representative Volume Element (RVE), Periodic Boundary Conditions (PBC), and associated them to the volume fraction (Vf) and the crack width as variables. The proposed microcapsule design is verified through computational fracture simulations.
Although it is impractical to avert subsequent natural disasters, advances in simulation science and seismological studies make it possible to lessen the catastrophic damage. There currently exists in many urban areas a large number of structures, which are prone to damage by earthquakes. These were constructed without the guidance of a national seismic code, either before it existed or before it was enforced. For instance, in Istanbul, Turkey, as a high seismic area, around 90% of buildings are substandard, which can be generalized into other earthquakeprone regions in Turkey. The reliability of this building stock resulting from earthquake-induced collapse is currently uncertain. Nonetheless, it is also not feasible to perform a detailed seismic vulnerability analysis on each building as a solution to the scenario, as it will be too complicated and expensive. This indicates the necessity of a reliable, rapid, and computationally easy method for seismic vulnerability assessment, commonly known as Rapid Visual Screening (RVS). In RVS methodology, an observational survey of buildings is performed, and according to the data collected during the visual inspection, a structural score is calculated without performing any structural calculations to determine the expected damage of a building and whether the building needs detailed assessment. Although this method might save time and resources due to the subjective/qualitative judgments of experts who performed the inspection, the evaluation process is dominated by vagueness and uncertainties, where the vagueness can be handled adequately through the fuzzy set theory but do not cover all sort of uncertainties due to its crisp membership functions. In this study, a novel method of rapid visual hazard safety assessment of buildings against earthquake is introduced in which an interval type-2 fuzzy logic system (IT2FLS) is used to cover uncertainties. In addition, the proposed method provides the possibility to evaluate the earthquake risk of the building by considering factors related to the building importance and exposure. A smartphone app prototype of the method has been introduced. For validation of the proposed method, two case studies have been selected, and the result of the analysis presents the robust efficiency of the proposed method.
In recent years increasingly consideration has been given to the lifetime extension of existing structures. This is based on the fact that a growing percentage of civil infrastructure as well as buildings is threatened by obsolescence and that due to simple monetary reasons this can no longer be countered by simply re-building everything anew. Hence maintenance interventions are required which allow partial or complete structural rehabilitation. However, maintenance interventions have to be economically reasonable, that is, maintenance expenditures have to be outweighed by expected future benefits. Is this not the case, then indeed the structure is obsolete - at least in its current functional, economic, technical, or social configuration - and innovative alternatives have to be evaluated. An optimization formulation for planning maintenance interventions based on cost-benefit criteria is proposed herein. The underlying formulation is as follows: (a) between maintenance interventions structural deterioration is described as a random process; (b) maintenance interventions can take place anytime throughout lifetime and comprise the rehabilitation of all deterioration states above a certain minimum level; and (c) maintenance interventions are optimized by taking into account all expected life-cycle costs (construction, failure, inspection and state-dependent repair costs) as well as state- or time-dependent benefit rates. The optimization is performed by an evolutionary algorithm. The proposed approach also allows to determine optimal lifetimes and acceptable failure rates. Numerical examples demonstrate the importance of defining benefit rates explicitly. It is shown, that the optimal solution to maintenance interventions requires to take action before reaching the acceptable failure rate or the zero expected net benefit rate level. Deferring decisions with respect to maintenance not only results, in general, in higher losses, but also results in overly hazardous structures.
This thesis addresses an adaptive higher-order method based on a Geometry Independent Field approximatTion(GIFT) of polynomial/rationals plines over hierarchical T-meshes(PHT/RHT-splines).
In isogeometric analysis, basis functions used for constructing geometric models in computer-aided design(CAD) are also employed to discretize the partial differential equations(PDEs) for numerical analysis. Non-uniform rational B-Splines(NURBS) are the most commonly used basis functions in CAD. However, they may not be ideal for numerical analysis where local refinement is required.
The alternative method GIFT deploys different splines for geometry and numerical analysis. NURBS are utilized for the geometry representation, while for the field solution, PHT/RHT-splines are used. PHT-splines not only inherit the useful properties of B-splines and NURBS, but also possess the capabilities of local refinement and hierarchical structure. The smooth basis function properties of PHT-splines make them suitable for analysis purposes. While most problems considered in isogeometric analysis can be solved efficiently when the solution is smooth, many non-trivial problems have rough solutions. For example, this can be caused by the presence of re-entrant corners in the domain. For such problems, a tensor-product basis (as in the case of NURBS) is less suitable for resolving the singularities that appear since refinement propagates throughout the computational domain. Hierarchical bases and local refinement (as in the case of PHT-splines) allow for a more efficient way to resolve these singularities by adding more degrees of freedom where they are necessary. In order to drive the adaptive refinement, an efficient recovery-based error estimator is proposed in this thesis. The estimator produces a recovery solution which is a more accurate approximation than the computed numerical solution. Several two- and three-dimensional numerical investigations with PHT-splines of higher order and continuity prove that the proposed method is capable of obtaining results with higher accuracy, better convergence, fewer degrees of freedom and less computational cost than NURBS for smooth solution problems. The adaptive GIFT method utilizing PHT-splines with the recovery-based error estimator is used for solutions with discontinuities or singularities where adaptive local refinement in particular domains of interest achieves higher accuracy with fewer degrees of freedom. This method also proves that it can handle complicated multi-patch domains for two- and three-dimensional problems outperforming uniform refinement in terms of degrees of freedom and computational cost.
Rechargeable lithium ion batteries (LIBs) play a very significant role in power supply and storage. In recent decades, LIBs have caught tremendous attention in mobile communication, portable electronics, and electric vehicles. Furthermore, global warming has become a worldwide issue due to the ongoing production of greenhouse gases. It motivates solutions such as renewable sources of energy. Solar and wind energies are the most important ones in renewable energy sources. By technology progress, they will definitely require batteries to store the produced power to make a balance between power generation and consumption. Nowadays,rechargeable batteries such as LIBs are considered as one of the best solutions. They provide high specific energy and high rate performance while their rate of self-discharge is low.
Performance of LIBs can be improved through the modification of battery characteristics. The size of solid particles in electrodes can impact the specific energy and the cyclability of batteries. It can improve the amount of lithium content in the electrode which is a vital parameter in capacity and capability of a battery. There exist diferent sources of heat generation in LIBs such as heat produced during electrochemical reactions, internal resistance in battery. The size of electrode's electroactive particles can directly affect the produced heat in battery. It will be shown that the smaller size of solid particle enhance the thermal characteristics of LIBs.
Thermal issues such as overheating, temperature maldistribution in the battery, and thermal runaway have confined applications of LIBs. Such thermal challenges reduce the Life cycle of LIBs. As well, they may lead to dangerous conditions such as fire or even explosion in batteries. However, recent advances in fabrication of advanced materials such as graphene and carbon nanotubes with extraordinary thermal conductivity and electrical properties propose new opportunities to enhance their performance. Since experimental works are expensive, our objective is to use computational methods to investigate the thermal issues in LIBS. Dissipation of the heat produced in the battery can improve the cyclability and specific capacity of LIBs. In real applications, packs of LIB consist several battery cells that are used as the power source. Therefore, it is worth to investigate thermal characteristic of battery packs under their cycles of charging/discharging operations at different applied current rates. To remove the produced heat in batteries, they can be surrounded by materials with high thermal conductivity. Parafin wax absorbs high energy since it has a high latent heat. Absorption high amounts of energy occurs at constant temperature without phase change. As well, thermal conductivity of parafin can be magnified with nano-materials such as graphene, CNT, and fullerene to form a nano-composite medium. Improving the thermal conductivity of LIBs increase the heat dissipation from batteries which is a vital issue in systems of battery thermal management. The application of two-dimensional (2D) materials has been on the rise since exfoliation the graphene from bulk graphite. 2D materials are single-layered in an order of nanosizes which show superior thermal, mechanical, and optoelectronic properties. They are potential candidates for energy storage and supply, particularly in lithium ion batteries as electrode material. The high thermal conductivity of graphene and graphene-like materials can play a significant role in thermal management of batteries. However, defects always exist in nano-materials since there is no ideal fabrication process. One of the most important defects in materials are nano-crack which can dramatically weaken the mechanical properties of the materials. Newly synthesized crystalline carbon nitride with the stoichiometry of C3N have attracted many attentions due to its extraordinary mechanical and thermal properties. The other nano-material is phagraphene which shows anisotropic mechanical characteristics which is ideal in production of nanocomposite.
It shows ductile fracture behavior when subjected under uniaxial loadings. It is worth to investigate their thermo-mechanical properties in its pristine and defective states. We hope that the findings of our work not only be useful for both experimental and theoretical researches but also help to design advanced electrodes for LIBs.
The importance of modern simulation methods in the mechanical analysis of heterogeneous solids is presented in detail. Thereby the problem is noted that even for small bodies the required high-resolution analysis reaches the limits of today's computational power, in terms of memory demand as well as acceptable computational effort. A further problem is that frequently the accuracy of geometrical modelling of heterogeneous bodies is inadequate. The present work introduces a systematic combination and adaption of grid-based methods for achieving an essentially higher resolution in the numerical analysis of heterogeneous solids. Grid-based methods are as well primely suited for developing efficient and numerically stable algorithms for flexible geometrical modeling. A key aspect is the uniform data management for a grid, which can be utilized to reduce the effort and complexity of almost all concerned methods. A new finite element program, called Mulgrido, was just developed to realize this concept consistently and to test the proposed methods. Several disadvantages which generally result from grid discretizations are selectively corrected by modified methods. The present work is structured into a geometrical model, a mechanical model and a numerical model. The geometrical model includes digital image-based modeling and in particular several methods for the theory-based generation of inclusion-matrix models. Essential contributions refer to variable shape, size distribution, separation checks and placement procedures of inclusions. The mechanical model prepares the fundamentals of continuum mechanics, homogenization and damage modeling for the following numerical methods. The first topic of the numerical model introduces to a special version of B-spline finite elements. These finite elements are entirely variable in the order k of B-splines. For homogeneous bodies this means that the approximation quality can arbitrarily be scaled. In addition, the multiphase finite element concept in combination with transition zones along material interfaces yields a valuable solution for heterogeneous bodies. As the formulation is element-based, the storage of a global stiffness matrix is superseded such that the memory demand can essentially be reduced. This is possible in combination with iterative solver methods which represent the second topic of the numerical model. Here, the focus lies on multigrid methods where the number of required operations to solve a linear equation system only increases linearly with problem size. Moreover, for badly conditioned problems quite an essential improvement is achieved by preconditioning. The third part of the numerical model discusses certain aspects of damage simulation which are closely related to the proposed grid discretization. The strong efficiency of the linear analysis can be maintained for damage simulation. This is achieved by a damage-controlled sequentially linear iteration scheme. Finally a study on the effective material behavior of heterogeneous bodies is presented. Especially the influence of inclusion shapes is examined. By means of altogether more than one hundred thousand random geometrical arrangements, the effective material behavior is statistically analyzed and assessed.
The computational costs of newly developed numerical simulation play a critical role in their acceptance within both academic use and industrial employment. Normally, the refinement of a method in the area of interest reduces the computational cost. This is unfortunately not true for most nonlocal simulation, since refinement typically increases the size of the material point neighborhood. Reducing the discretization size while keep- ing the neighborhood size will often require extra consideration. Peridy- namic (PD) is a newly developed numerical method with nonlocal nature. Its straightforward integral form equation of motion allows simulating dy- namic problems without any extra consideration required. The formation of crack and its propagation is known as natural to peridynamic. This means that discontinuity is a result of the simulation and does not demand any post-processing. As with other nonlocal methods, PD is considered an expensive method. The refinement of the nodal spacing while keeping the neighborhood size (i.e., horizon radius) constant, emerges to several nonphysical phenomena.
This research aims to reduce the peridynamic computational and imple- mentation costs. A novel refinement approach is introduced. The pro- posed approach takes advantage of the PD flexibility in choosing the shape of the horizon by introducing multiple domains (with no intersections) to the nodes of the refinement zone. It will be shown that no ghost forces will be created when changing the horizon sizes in both subdomains. The approach is applied to both bond-based and state-based peridynamic and verified for a simple wave propagation refinement problem illustrating the efficiency of the method. Further development of the method for higher dimensions proves to have a direct relationship with the mesh sensitivity of the PD. A method for solving the mesh sensitivity of the PD is intro- duced. The application of the method will be examined by solving a crack propagation problem similar to those reported in the literature.
New software architecture is proposed considering both academic and in- dustrial use. The available simulation tools for employing PD will be collected, and their advantages and drawbacks will be addressed. The challenges of implementing any node base nonlocal methods while max- imizing the software flexibility to further development and modification
will be discussed and addressed. A software named Relation-Based Sim- ulator (RBS) is developed for examining the proposed architecture. The exceptional capabilities of RBS will be explored by simulating three dis- tinguished models. RBS is available publicly and open to further develop- ment. The industrial acceptance of the RBS will be tested by targeting its performance on one Mac and two Linux distributions.
The computational costs of newly developed numerical simulation play a critical role in their acceptance within both academic use and industrial employment. Normally, the refinement of a method in the area of interest reduces the computational cost. This is unfortunately not true for most nonlocal simulation, since refinement typically increases the size of the material point neighborhood. Reducing the discretization size while keep- ing the neighborhood size will often require extra consideration. Peridynamic (PD) is a newly developed numerical method with nonlocal nature. Its straightforward integral form equation of motion allows simulating dynamic problems without any extra consideration required. The formation of crack and its propagation is known as natural to peridynamic. This means that discontinuity is a result of the simulation and does not demand any post-processing. As with other nonlocal methods, PD is considered an expensive method. The refinement of the nodal spacing while keeping the neighborhood size (i.e., horizon radius) constant, emerges to several nonphysical phenomena.
This research aims to reduce the peridynamic computational and imple- mentation costs. A novel refinement approach is introduced. The pro- posed approach takes advantage of the PD flexibility in choosing the shape of the horizon by introducing multiple domains (with no intersections) to the nodes of the refinement zone. It will be shown that no ghost forces will be created when changing the horizon sizes in both subdomains. The approach is applied to both bond-based and state-based peridynamic and verified for a simple wave propagation refinement problem illustrating the efficiency of the method. Further development of the method for higher dimensions proves to have a direct relationship with the mesh sensitivity of the PD. A method for solving the mesh sensitivity of the PD is intro- duced. The application of the method will be examined by solving a crack propagation problem similar to those reported in the literature.
New software architecture is proposed considering both academic and in- dustrial use. The available simulation tools for employing PD will be collected, and their advantages and drawbacks will be addressed. The challenges of implementing any node base nonlocal methods while max- imizing the software flexibility to further development and modification will be discussed and addressed. A software named Relation-Based Sim- ulator (RBS) is developed for examining the proposed architecture. The exceptional capabilities of RBS will be explored by simulating three distinguished models. RBS is available publicly and open to further develop- ment. The industrial acceptance of the RBS will be tested by targeting its performance on one Mac and two Linux distributions.
Methods based on B-splines for model representation, numerical analysis and image registration
(2015)
The thesis consists of inter-connected parts for modeling and analysis using newly developed isogeometric methods. The main parts are reproducing kernel triangular B-splines, extended isogeometric analysis for solving weakly discontinuous problems, collocation methods using superconvergent points, and B-spline basis in image registration applications.
Each topic is oriented towards application of isogeometric analysis basis functions to ease the process of integrating the modeling and analysis phases of simulation.
First, we develop reproducing a kernel triangular B-spline-based FEM for solving PDEs. We review the triangular B-splines and their properties. By definition, the triangular basis function is very flexible in modeling complicated domains. However, instability results when it is applied for analysis. We modify the triangular B-spline by a reproducing kernel technique, calculating a correction term for the triangular kernel function from the chosen surrounding basis. The improved triangular basis is capable to obtain the results with higher accuracy and almost optimal convergence rates.
Second, we propose an extended isogeometric analysis for dealing with weakly discontinuous problems such as material interfaces. The original IGA is combined with XFEM-like enrichments which are continuous functions themselves but with discontinuous derivatives. Consequently, the resulting solution space can approximate solutions with weak discontinuities. The method is also applied to curved material interfaces, where the inverse mapping and the curved triangular elements are considered.
Third, we develop an IGA collocation method using superconvergent points. The collocation methods are efficient because no numerical integration is needed. In particular when higher polynomial basis applied, the method has a lower computational cost than Galerkin methods. However, the positions of the collocation points are crucial for the accuracy of the method, as they affect the convergent rate significantly. The proposed IGA collocation method uses superconvergent points instead of the traditional Greville abscissae points. The numerical results show the proposed method can have better accuracy and optimal convergence rates, while the traditional IGA collocation has optimal convergence only for even polynomial degrees.
Lastly, we propose a novel dynamic multilevel technique for handling image registration. It is application of the B-spline functions in image processing. The procedure considered aims to align a target image from a reference image by a spatial transformation. The method starts with an energy function which is the same as a FEM-based image registration. However, we simplify the solving procedure, working on the energy function directly. We dynamically solve for control points which are coefficients of B-spline basis functions. The new approach is more simple and fast. Moreover, it is also enhanced by a multilevel technique in order to prevent instabilities. The numerical testing consists of two artificial images, four real bio-medical MRI brain and CT heart images, and they show our registration method is accurate, fast and efficient, especially for large deformation problems.
Matrix-free voxel-based finite element method for materials with heterogeneous microstructures
(2019)
Modern image detection techniques such as micro computer tomography
(μCT), magnetic resonance imaging (MRI) and scanning electron microscopy (SEM) provide us with high resolution images of the microstructure of materials in a non-invasive and convenient way. They form the basis for the geometrical models of high-resolution analysis, so called image-based analysis.
However especially in 3D, discretizations of these models reach easily the size of 100 Mill. degrees of freedoms and require extensive hardware resources in terms of main memory and computing power to solve the numerical model. Consequently, the focus of this work is to combine and adapt numerical solution methods to reduce the memory demand first and then the computation time and therewith enable an execution of the image-based analysis on modern computer desktops. Hence, the numerical model is a straightforward grid discretization of the voxel-based (pixels with a third dimension) geometry which omits the boundary detection algorithms and allows reduced storage of the finite element data structure and a matrix-free solution algorithm.
This in turn reduce the effort of almost all applied grid-based solution techniques and results in memory efficient and numerically stable algorithms for the microstructural models. Two variants of the matrix-free algorithm are presented. The efficient iterative solution method of conjugate gradients is used with matrix-free applicable preconditioners such as the Jacobi and the especially suited multigrid method. The jagged material boundaries of the voxel-based mesh are smoothed through embedded boundary elements which contain different material information at the integration point and are integrated sub-cell wise though without additional boundary detection. The efficiency of the matrix-free methods can be retained.
In recent years, lightweight materials, such as polymer composite materials (PNCs) have been studied and developed due to their excellent physical and chemical properties. Structures composed of these composite materials are widely used in aerospace engineering structures, automotive components, and electrical devices. The excellent and outstanding mechanical, thermal, and electrical properties of Carbon nanotube (CNT) make it an ideal filler to strengthen polymer materials’ comparable properties. The heat transfer of composite materials has very promising engineering applications in many fields, especially in electronic devices and energy storage equipment. It is essential in high-energy density systems since electronic components need heat dissipation functionality. Or in other words, in electronic devices the generated heat should ideally be dissipated by light and small heat sinks.
Polymeric composites consist of fillers embedded in a polymer matrix, the first ones will significantly affect the overall (macroscopic) performance of the material. There are many common carbon-based fillers such as single-walled carbon nanotubes (SWCNT), multi-walled carbon nanotubes (MWCNT), carbon nanobuds (CNB), fullerene, and graphene. Additives inside the matrix have become a popular subject for researchers. Some extraordinary characters, such as high-performance load, lightweight design, excellent chemical resistance, easy processing, and heat transfer, make the design of polymeric nanotube composites (PNCs) flexible. Due to the reinforcing effects with different fillers on composite materials, it has a higher degree of freedom and can be designed for the structure according to specific applications’ needs. As already stated, our research focus will be on SWCNT enhanced PNCs. Since experiments are timeconsuming, sometimes expensive and cannot shed light into phenomena taking place for instance at the interfaces/interphases of composites, they are often complemented through theoretical and computational analysis.
While most studies are based on deterministic approaches, there is a comparatively lower number of stochastic methods accounting for uncertainties in the input parameters. In deterministic models, the output of the model is fully determined by the parameter values and the initial conditions. However, uncertainties in the input parameters such as aspect ratio, volume fraction, thermal properties of fiber and matrix need to be taken into account for reliable predictions. In this research, a stochastic multiscale method is provided to study the influence of numerous uncertain input parameters on the thermal conductivity of the composite. Therefore, a hierarchical multi-scale method based on computational homogenization is presented in to predict the macroscopic thermal conductivity based on the fine-scale structure. In order to study the inner mechanism, we use the finite element method and employ surrogate models to conduct a Global Sensitivity Analysis (GSA). The SA is performed in order to quantify the influence of the conductivity of the fiber, matrix, Kapitza resistance, volume fraction and aspect ratio on the macroscopic conductivity. Therefore, we compute first-order and total-effect sensitivity indices with different surrogate models.
As stochastic multiscale models are computational expensive, surrogate approaches are commonly exploited. With the emergence of high performance computing and artificial intelligence, machine learning has become a popular modeling tool for numerous applications. Machine learning (ML) is commonly used in regression and maps data through specific rules with algorithms to build input and output models. They are particularly useful for nonlinear input-output relationships when sufficient data is available. ML has also been used in the design of new materials and multiscale analysis. For instance, Artificial neural networks and integrated learning seem to be ideally for such a task. They can theoretically simulate any non-linear relationship through the connection of neurons. Mapping relationships are employed to carry out data-driven simulations of inputs and outputs in stochastic modeling.
This research aims to develop a stochastic multi-scale computational models of PNCs in heat transfer. Multi-scale stochastic modeling with uncertainty analysis and machine learning methods consist of the following components:
-Uncertainty Analysis. A surrogate based global sensitivity analysis is coupled with a hierarchical multi-scale method employing computational homogenization. The effect of the conductivity of the fibers and the matrix, the Kapitza resistance, volume fraction and aspect ratio on the ’macroscopic’ conductivity of the composite is systematically studied. All selected surrogate models yield consistently the conclusions that the most influential input parameters are the aspect ratio followed by the volume fraction. The Kapitza Resistance has no significant effect on the thermal conductivity of the PNCs. The most accurate surrogate model in terms of the R2 value is the moving least square (MLS).
-Hybrid Machine Learning Algorithms. A combination of artificial neural network (ANN) and particle swarm optimization (PSO) is applied to estimate the relationship between variable input and output parameters. The ANN is used for modeling the composite while PSO improves the prediction performance through an optimized global minimum search. The thermal conductivity of the fibers and the matrix, the kapitza resistance, volume fraction and aspect ratio are selected as input parameters. The output is the macroscopic (homogenized) thermal conductivity of the composite. The results show that the PSO significantly improves the predictive ability of this hybrid intelligent algorithm, which outperforms traditional neural networks.
-Stochastic Integrated Machine Learning. A stochastic integrated machine learning based multiscale approach for the prediction of the macroscopic thermal conductivity in PNCs is developed. Seven types of machine learning models are exploited in this research, namely Multivariate Adaptive Regression Splines (MARS), Support Vector Machine (SVM), Regression Tree (RT), Bagging Tree (Bag), Random Forest (RF), Gradient Boosting Machine (GBM) and Cubist. They are used as components of stochastic modeling to construct the relationship between the variable of the inputs’ uncertainty and the macroscopic thermal conductivity of PNCs. Particle Swarm Optimization (PSO) is used for hyper-parameter tuning to find the global optimal values leading to a significant reduction in the computational cost. The advantages and disadvantages of various methods are also analyzed in terms of computing time and model complexity to finally give a recommendation for the applicability of different models.
The numerical simulation of damage using phenomenological models on the macroscale was state of the art for many decades. However, such models are not able to capture the complex nature of damage, which simultaneously proceeds on multiple length scales. Furthermore, these phenomenological models usually contain damage parameters, which are physically not interpretable. Consequently, a reasonable experimental determination of these parameters is often impossible. In the last twenty years, the ongoing advance in computational capacities provided new opportunities for more and more detailed studies of the microstructural damage behavior. Today, multiphase models with several million degrees of freedom enable for the numerical simulation of micro-damage phenomena in naturally heterogeneous materials. Therewith, the application of multiscale concepts for the numerical investigation of the complex nature of damage can be realized. The presented thesis contributes to a hierarchical multiscale strategy for the simulation of brittle intergranular damage in polycrystalline materials, for example aluminum. The numerical investigation of physical damage phenomena on an atomistic microscale and the integration of these physically based information into damage models on the continuum meso- and macroscale is intended. Therefore, numerical methods for the damage analysis on the micro- and mesoscale including the scale transfer are presented and the transition to the macroscale is discussed. The investigation of brittle intergranular damage on the microscale is realized by the application of the nonlocal Quasicontinuum method, which fully describes the material behavior by atomistic potential functions, but reduces the number of atomic degrees of freedom by introducing kinematic couplings. Since this promising method is applied only by a limited group of researchers for special problems, necessary improvements have been realized in an own parallelized implementation of the 3D nonlocal Quasicontinuum method. The aim of this implementation was to develop and combine robust and efficient algorithms for a general use of the Quasicontinuum method, and therewith to allow for the atomistic damage analysis in arbitrary grain boundary configurations. The implementation is applied in analyses of brittle intergranular damage in ideal and nonideal grain boundary models of FCC aluminum, considering arbitrary misorientations. From the microscale simulations traction separation laws are derived, which describe grain boundary decohesion on the mesoscale. Traction separation laws are part of cohesive zone models to simulate the brittle interface decohesion in heterogeneous polycrystal structures. 2D and 3D mesoscale models are presented, which are able to reproduce crack initiation and propagation along cohesive interfaces in polycrystals. An improved Voronoi algorithm is developed in 2D to generate polycrystal material structures based on arbitrary distribution functions of grain size. The new model is more flexible in representing realistic grain size distributions. Further improvements of the 2D model are realized by the implementation and application of an orthotropic material model with Hill plasticity criterion to grains. The 2D and 3D polycrystal models are applied to analyze crack initiation and propagation in statically loaded samples of aluminum on the mesoscale without the necessity of initial damage definition.
Structural vibration control of high-speed railway bridges using tuned mass dampers, semi-active tuned mass dampers, fluid viscous dampers and magnetorheological dampers to reduce resonant structural vibrations is studied. In this work, the addressed main issues include modeling of the dynamic interaction of the structures, optimization of the parameters of the dampers and comparison of their efficiency.
A new approach to optimize multiple tuned mass damper systems on an uncertain model is proposed based on the H-infinity optimization criteria and the DK iteration procedure with norm-bounded uncertainties in frequency domain. The parameters of tuned mass dampers are optimized directly and simultaneously on different modes contributing significantly to the multi-resonant peaks to explore the different possible combinations of parameters. The effectiveness of the present method is also evaluated through comparison with a previous method.
In the case of semi-active tuned mass dampers, an optimization algorithm is derived to control the magnetorheological damper in these semi-active damping systems. The use of the proposed algorithm can generate various combinations of control gains and state variables. This can lead to the improvement of the ability of MR dampers to track the desired control forces. An uncertain model to reduce detuning effects is also considered in this work.
Next, for fluid viscous dampers, in order to tune the optimal parameters of fluid viscous dampers to the vicinity of the exact values, analytical formulae which can include structural damping are developed based on the perturbation method. The proposed formulae can also be considered as an improvement of the previous analytical formulae, especially for bridge beams with large structural damping.
Finally, a new combination of magnetorheological dampers and a double-beam system to improve the performance of the primary structure vibration is proposed. An algorithm to control magnetorheological dampers in this system is developed by using standard linear matrix inequality techniques. Weight functions as a loop shaping procedure are also introduced in the feedback controllers to improve the tracking ability of magnetorheological damping forces. To this end, the effectiveness of magnetorheological dampers controlled by the proposed scheme, along with the effects of the uncertain and time-delay parameters on the models, are evaluated through numerical simulations.
Additionally, a comparison of the dampers based on their performance is also considered in this work.
Encapsulation-based self-healing concrete has received a lot of attention nowadays in civil engineering field. These capsules are embedded in the cementitious matrix during concrete mixing. When the cracks appear, the embedded capsules which are placed along the path of incoming crack are fractured and then release of healing agents in the vicinity of damage. The materials of capsules need to be designed in a way that they should be able to break with small deformation, so the internal fluid can be released to seal the crack. This study focuses on computational modeling of fracture in encapsulation-based selfhealing concrete. The numerical model of 2D and 3D with randomly packed aggreates and capsules have been developed to analyze fracture mechanism that plays a significant role in the fracture probability of capsules and consequently the self-healing process. The capsules are assumed to be made of Poly Methyl Methacrylate (PMMA) and the potential cracks are represented by pre-inserted cohesive elements with tension and shear softening laws along the element boundaries of the mortar matrix, aggregates, capsules, and at the interfaces between these phases. The effects of volume fraction, core-wall thickness ratio, and mismatch fracture properties of capsules on the load carrying capacity of self-healing concrete and fracture probability of the capsules are investigated. The output of this study will become valuable tool to assist not only the experimentalists but also the manufacturers in designing an appropriate capsule material for self-healing concrete.
Phase Field Modeling for Fracture with Applications to Homogeneous and Heterogeneous Materials
(2017)
The thesis presents an implementation including different applications of a variational-based approach for gradient type standard dissipative solids. Phase field model for brittle fracture is an application of the variational-based framework for gradient type solids. This model allows the prediction of different crack topologies and states. Of significant concern is the application of theoretical and numerical formulation of the phase field modeling into the commercial finite element software Abaqus in 2D and 3D. The fully coupled incremental variational formulation of phase field method is implemented by using the UEL and UMAT subroutines of Abaqus. The phase field method
considerably reduces the implementation complexity of fracture problems as it removes the need for numerical tracking of discontinuities in the displacement field that are characteristic of discrete crack methods. This is accomplished by replacing the sharp discontinuities with a scalar damage phase field representing the diffuse crack topology wherein the amount of diffusion is controlled by a regularization parameter. The nonlinear coupled system consisting of the linear momentum equation and a diffusion type equation governing the phase field evolution is solved simultaneously via a Newton-
Raphson approach. Post-processing of simulation results to be used as visualization
module is performed via an additional UMAT subroutine implemented in the standard Abaqus viewer.
In the same context, we propose a simple yet effective algorithm to initiate and propagate cracks in 2D geometries which is independent of both particular constitutive laws and specific element technology and dimension. It consists of a localization limiter in the form of the screened Poisson equation with, optionally, local mesh refinement. A staggered scheme for standard equilibrium and screened Cauchy equations is used. The remeshing part of the algorithm consists of a sequence of mesh subdivision and element erosion steps. Element subdivision is based on edge split operations using a
given constitutive quantity (either damage or void fraction). Mesh smoothing makes use of edge contraction as function of a given constitutive quantity such as the principal stress or void fraction. To assess the robustness and accuracy of this algorithm, we use both quasi-brittle benchmarks and ductile tests.
Furthermore, we introduce a computational approach regarding mechanical loading in microscale on an inelastically deforming composite material. The nanocomposites material of fully exfoliated clay/epoxy is shaped to predict macroscopic elastic and fracture related material parameters based on their fine–scale features. Two different configurations of polymer nanocomposites material (PNCs) have been studied. These configurations are fully bonded PNCs and PNCs with an interphase zone formation between the matrix and the clay reinforcement. The representative volume element of PNCs specimens with different clay weight contents, different aspect ratios, and different
interphase zone thicknesses are generated by adopting Python scripting. Different constitutive models are employed for the matrix, the clay platelets, and the interphase zones. The brittle fracture behavior of the epoxy matrix and the interphase zones material are modeled using the phase field approach, whereas the stiff silicate clay platelets of the composite are designated as a linear elastic material. The comprehensive study investigates the elastic and fracture behavior of PNCs composites, in addition to predict Young’s modulus, tensile strength, fracture toughness, surface energy dissipation, and cracks surface area in the composite for different material parameters, geometry, and interphase zones properties and thicknesses.